Properties

Label 3840.2.a.bj.1.2
Level $3840$
Weight $2$
Character 3840.1
Self dual yes
Analytic conductor $30.663$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(1,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.6625543762\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +4.82843 q^{11} -4.82843 q^{13} -1.00000 q^{15} -2.82843 q^{17} -2.00000 q^{21} +5.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.65685 q^{29} -6.82843 q^{31} +4.82843 q^{33} +2.00000 q^{35} +10.4853 q^{37} -4.82843 q^{39} +7.65685 q^{41} -9.65685 q^{43} -1.00000 q^{45} -9.65685 q^{47} -3.00000 q^{49} -2.82843 q^{51} -0.343146 q^{53} -4.82843 q^{55} -0.828427 q^{59} -1.65685 q^{61} -2.00000 q^{63} +4.82843 q^{65} +1.65685 q^{67} +5.65685 q^{69} -2.34315 q^{71} -13.3137 q^{73} +1.00000 q^{75} -9.65685 q^{77} +4.48528 q^{79} +1.00000 q^{81} +2.82843 q^{85} -7.65685 q^{87} -15.6569 q^{89} +9.65685 q^{91} -6.82843 q^{93} -3.65685 q^{97} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} - 2 q^{15} - 4 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 8 q^{31} + 4 q^{33} + 4 q^{35} + 4 q^{37} - 4 q^{39} + 4 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} - 6 q^{49} - 12 q^{53} - 4 q^{55} + 4 q^{59} + 8 q^{61} - 4 q^{63} + 4 q^{65} - 8 q^{67} - 16 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{87} - 20 q^{89} + 8 q^{91} - 8 q^{93} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 0 0
\(13\) −4.82843 −1.33916 −0.669582 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 0 0
\(33\) 4.82843 0.840521
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 10.4853 1.72377 0.861885 0.507104i \(-0.169284\pi\)
0.861885 + 0.507104i \(0.169284\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −9.65685 −1.40860 −0.704298 0.709904i \(-0.748738\pi\)
−0.704298 + 0.709904i \(0.748738\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) −0.343146 −0.0471347 −0.0235673 0.999722i \(-0.507502\pi\)
−0.0235673 + 0.999722i \(0.507502\pi\)
\(54\) 0 0
\(55\) −4.82843 −0.651065
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.828427 −0.107852 −0.0539260 0.998545i \(-0.517174\pi\)
−0.0539260 + 0.998545i \(0.517174\pi\)
\(60\) 0 0
\(61\) −1.65685 −0.212138 −0.106069 0.994359i \(-0.533827\pi\)
−0.106069 + 0.994359i \(0.533827\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 4.82843 0.598893
\(66\) 0 0
\(67\) 1.65685 0.202417 0.101208 0.994865i \(-0.467729\pi\)
0.101208 + 0.994865i \(0.467729\pi\)
\(68\) 0 0
\(69\) 5.65685 0.681005
\(70\) 0 0
\(71\) −2.34315 −0.278080 −0.139040 0.990287i \(-0.544402\pi\)
−0.139040 + 0.990287i \(0.544402\pi\)
\(72\) 0 0
\(73\) −13.3137 −1.55825 −0.779126 0.626868i \(-0.784337\pi\)
−0.779126 + 0.626868i \(0.784337\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −9.65685 −1.10050
\(78\) 0 0
\(79\) 4.48528 0.504634 0.252317 0.967645i \(-0.418807\pi\)
0.252317 + 0.967645i \(0.418807\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) −7.65685 −0.820901
\(88\) 0 0
\(89\) −15.6569 −1.65962 −0.829812 0.558044i \(-0.811552\pi\)
−0.829812 + 0.558044i \(0.811552\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) 0 0
\(93\) −6.82843 −0.708075
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 0 0
\(99\) 4.82843 0.485275
\(100\) 0 0
\(101\) 17.3137 1.72278 0.861389 0.507946i \(-0.169595\pi\)
0.861389 + 0.507946i \(0.169595\pi\)
\(102\) 0 0
\(103\) −15.6569 −1.54272 −0.771358 0.636402i \(-0.780422\pi\)
−0.771358 + 0.636402i \(0.780422\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) −13.6569 −1.32026 −0.660129 0.751152i \(-0.729498\pi\)
−0.660129 + 0.751152i \(0.729498\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 10.4853 0.995219
\(112\) 0 0
\(113\) −3.51472 −0.330637 −0.165318 0.986240i \(-0.552865\pi\)
−0.165318 + 0.986240i \(0.552865\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) 0 0
\(117\) −4.82843 −0.446388
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 7.65685 0.690395
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.34315 −0.385392 −0.192696 0.981259i \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(128\) 0 0
\(129\) −9.65685 −0.850239
\(130\) 0 0
\(131\) −2.48528 −0.217140 −0.108570 0.994089i \(-0.534627\pi\)
−0.108570 + 0.994089i \(0.534627\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 6.82843 0.583392 0.291696 0.956511i \(-0.405780\pi\)
0.291696 + 0.956511i \(0.405780\pi\)
\(138\) 0 0
\(139\) 1.65685 0.140533 0.0702663 0.997528i \(-0.477615\pi\)
0.0702663 + 0.997528i \(0.477615\pi\)
\(140\) 0 0
\(141\) −9.65685 −0.813254
\(142\) 0 0
\(143\) −23.3137 −1.94959
\(144\) 0 0
\(145\) 7.65685 0.635867
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) −0.343146 −0.0281116 −0.0140558 0.999901i \(-0.504474\pi\)
−0.0140558 + 0.999901i \(0.504474\pi\)
\(150\) 0 0
\(151\) −8.48528 −0.690522 −0.345261 0.938507i \(-0.612210\pi\)
−0.345261 + 0.938507i \(0.612210\pi\)
\(152\) 0 0
\(153\) −2.82843 −0.228665
\(154\) 0 0
\(155\) 6.82843 0.548472
\(156\) 0 0
\(157\) 5.51472 0.440122 0.220061 0.975486i \(-0.429374\pi\)
0.220061 + 0.975486i \(0.429374\pi\)
\(158\) 0 0
\(159\) −0.343146 −0.0272132
\(160\) 0 0
\(161\) −11.3137 −0.891645
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) −4.82843 −0.375893
\(166\) 0 0
\(167\) −10.3431 −0.800377 −0.400188 0.916433i \(-0.631055\pi\)
−0.400188 + 0.916433i \(0.631055\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.6569 −1.19037 −0.595184 0.803589i \(-0.702921\pi\)
−0.595184 + 0.803589i \(0.702921\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) −0.828427 −0.0622684
\(178\) 0 0
\(179\) 2.48528 0.185759 0.0928793 0.995677i \(-0.470393\pi\)
0.0928793 + 0.995677i \(0.470393\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) −1.65685 −0.122478
\(184\) 0 0
\(185\) −10.4853 −0.770893
\(186\) 0 0
\(187\) −13.6569 −0.998688
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −27.3137 −1.97635 −0.988175 0.153328i \(-0.951001\pi\)
−0.988175 + 0.153328i \(0.951001\pi\)
\(192\) 0 0
\(193\) 22.9706 1.65346 0.826729 0.562601i \(-0.190199\pi\)
0.826729 + 0.562601i \(0.190199\pi\)
\(194\) 0 0
\(195\) 4.82843 0.345771
\(196\) 0 0
\(197\) −21.3137 −1.51854 −0.759269 0.650776i \(-0.774444\pi\)
−0.759269 + 0.650776i \(0.774444\pi\)
\(198\) 0 0
\(199\) −18.8284 −1.33471 −0.667356 0.744739i \(-0.732574\pi\)
−0.667356 + 0.744739i \(0.732574\pi\)
\(200\) 0 0
\(201\) 1.65685 0.116865
\(202\) 0 0
\(203\) 15.3137 1.07481
\(204\) 0 0
\(205\) −7.65685 −0.534778
\(206\) 0 0
\(207\) 5.65685 0.393179
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 25.6569 1.76629 0.883145 0.469099i \(-0.155421\pi\)
0.883145 + 0.469099i \(0.155421\pi\)
\(212\) 0 0
\(213\) −2.34315 −0.160550
\(214\) 0 0
\(215\) 9.65685 0.658592
\(216\) 0 0
\(217\) 13.6569 0.927088
\(218\) 0 0
\(219\) −13.3137 −0.899657
\(220\) 0 0
\(221\) 13.6569 0.918659
\(222\) 0 0
\(223\) 14.9706 1.00250 0.501252 0.865302i \(-0.332873\pi\)
0.501252 + 0.865302i \(0.332873\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.34315 −0.421009 −0.210505 0.977593i \(-0.567511\pi\)
−0.210505 + 0.977593i \(0.567511\pi\)
\(228\) 0 0
\(229\) 10.3431 0.683494 0.341747 0.939792i \(-0.388981\pi\)
0.341747 + 0.939792i \(0.388981\pi\)
\(230\) 0 0
\(231\) −9.65685 −0.635374
\(232\) 0 0
\(233\) −27.7990 −1.82117 −0.910586 0.413319i \(-0.864369\pi\)
−0.910586 + 0.413319i \(0.864369\pi\)
\(234\) 0 0
\(235\) 9.65685 0.629944
\(236\) 0 0
\(237\) 4.48528 0.291350
\(238\) 0 0
\(239\) −22.6274 −1.46365 −0.731823 0.681495i \(-0.761330\pi\)
−0.731823 + 0.681495i \(0.761330\pi\)
\(240\) 0 0
\(241\) −24.6274 −1.58639 −0.793196 0.608967i \(-0.791584\pi\)
−0.793196 + 0.608967i \(0.791584\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.17157 0.200188 0.100094 0.994978i \(-0.468086\pi\)
0.100094 + 0.994978i \(0.468086\pi\)
\(252\) 0 0
\(253\) 27.3137 1.71720
\(254\) 0 0
\(255\) 2.82843 0.177123
\(256\) 0 0
\(257\) 9.17157 0.572107 0.286053 0.958214i \(-0.407656\pi\)
0.286053 + 0.958214i \(0.407656\pi\)
\(258\) 0 0
\(259\) −20.9706 −1.30305
\(260\) 0 0
\(261\) −7.65685 −0.473947
\(262\) 0 0
\(263\) −18.3431 −1.13109 −0.565543 0.824719i \(-0.691334\pi\)
−0.565543 + 0.824719i \(0.691334\pi\)
\(264\) 0 0
\(265\) 0.343146 0.0210793
\(266\) 0 0
\(267\) −15.6569 −0.958184
\(268\) 0 0
\(269\) 3.65685 0.222962 0.111481 0.993767i \(-0.464441\pi\)
0.111481 + 0.993767i \(0.464441\pi\)
\(270\) 0 0
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) 0 0
\(273\) 9.65685 0.584459
\(274\) 0 0
\(275\) 4.82843 0.291165
\(276\) 0 0
\(277\) 28.8284 1.73213 0.866066 0.499929i \(-0.166641\pi\)
0.866066 + 0.499929i \(0.166641\pi\)
\(278\) 0 0
\(279\) −6.82843 −0.408807
\(280\) 0 0
\(281\) 22.9706 1.37031 0.685154 0.728398i \(-0.259735\pi\)
0.685154 + 0.728398i \(0.259735\pi\)
\(282\) 0 0
\(283\) −20.9706 −1.24657 −0.623285 0.781995i \(-0.714202\pi\)
−0.623285 + 0.781995i \(0.714202\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −15.3137 −0.903940
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −3.65685 −0.214369
\(292\) 0 0
\(293\) −5.31371 −0.310430 −0.155215 0.987881i \(-0.549607\pi\)
−0.155215 + 0.987881i \(0.549607\pi\)
\(294\) 0 0
\(295\) 0.828427 0.0482329
\(296\) 0 0
\(297\) 4.82843 0.280174
\(298\) 0 0
\(299\) −27.3137 −1.57959
\(300\) 0 0
\(301\) 19.3137 1.11322
\(302\) 0 0
\(303\) 17.3137 0.994647
\(304\) 0 0
\(305\) 1.65685 0.0948712
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −15.6569 −0.890687
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) −10.9706 −0.620093 −0.310046 0.950721i \(-0.600345\pi\)
−0.310046 + 0.950721i \(0.600345\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 9.31371 0.523110 0.261555 0.965189i \(-0.415765\pi\)
0.261555 + 0.965189i \(0.415765\pi\)
\(318\) 0 0
\(319\) −36.9706 −2.06995
\(320\) 0 0
\(321\) −13.6569 −0.762251
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.82843 −0.267833
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 19.3137 1.06480
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) 10.4853 0.574590
\(334\) 0 0
\(335\) −1.65685 −0.0905236
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −3.51472 −0.190893
\(340\) 0 0
\(341\) −32.9706 −1.78546
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −5.65685 −0.304555
\(346\) 0 0
\(347\) 1.65685 0.0889446 0.0444723 0.999011i \(-0.485839\pi\)
0.0444723 + 0.999011i \(0.485839\pi\)
\(348\) 0 0
\(349\) 36.9706 1.97899 0.989494 0.144571i \(-0.0461802\pi\)
0.989494 + 0.144571i \(0.0461802\pi\)
\(350\) 0 0
\(351\) −4.82843 −0.257722
\(352\) 0 0
\(353\) 25.4558 1.35488 0.677439 0.735579i \(-0.263090\pi\)
0.677439 + 0.735579i \(0.263090\pi\)
\(354\) 0 0
\(355\) 2.34315 0.124361
\(356\) 0 0
\(357\) 5.65685 0.299392
\(358\) 0 0
\(359\) 16.9706 0.895672 0.447836 0.894116i \(-0.352195\pi\)
0.447836 + 0.894116i \(0.352195\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 12.3137 0.646302
\(364\) 0 0
\(365\) 13.3137 0.696871
\(366\) 0 0
\(367\) 25.3137 1.32136 0.660682 0.750666i \(-0.270267\pi\)
0.660682 + 0.750666i \(0.270267\pi\)
\(368\) 0 0
\(369\) 7.65685 0.398600
\(370\) 0 0
\(371\) 0.686292 0.0356305
\(372\) 0 0
\(373\) 13.5147 0.699766 0.349883 0.936793i \(-0.386221\pi\)
0.349883 + 0.936793i \(0.386221\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 36.9706 1.90408
\(378\) 0 0
\(379\) 32.2843 1.65833 0.829166 0.559003i \(-0.188816\pi\)
0.829166 + 0.559003i \(0.188816\pi\)
\(380\) 0 0
\(381\) −4.34315 −0.222506
\(382\) 0 0
\(383\) −17.6569 −0.902223 −0.451112 0.892468i \(-0.648972\pi\)
−0.451112 + 0.892468i \(0.648972\pi\)
\(384\) 0 0
\(385\) 9.65685 0.492159
\(386\) 0 0
\(387\) −9.65685 −0.490885
\(388\) 0 0
\(389\) −9.31371 −0.472224 −0.236112 0.971726i \(-0.575873\pi\)
−0.236112 + 0.971726i \(0.575873\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −2.48528 −0.125366
\(394\) 0 0
\(395\) −4.48528 −0.225679
\(396\) 0 0
\(397\) 10.4853 0.526241 0.263121 0.964763i \(-0.415248\pi\)
0.263121 + 0.964763i \(0.415248\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.62742 −0.430833 −0.215416 0.976522i \(-0.569111\pi\)
−0.215416 + 0.976522i \(0.569111\pi\)
\(402\) 0 0
\(403\) 32.9706 1.64238
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 50.6274 2.50951
\(408\) 0 0
\(409\) −25.3137 −1.25168 −0.625841 0.779951i \(-0.715244\pi\)
−0.625841 + 0.779951i \(0.715244\pi\)
\(410\) 0 0
\(411\) 6.82843 0.336821
\(412\) 0 0
\(413\) 1.65685 0.0815285
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.65685 0.0811365
\(418\) 0 0
\(419\) 14.4853 0.707652 0.353826 0.935311i \(-0.384880\pi\)
0.353826 + 0.935311i \(0.384880\pi\)
\(420\) 0 0
\(421\) −16.9706 −0.827095 −0.413547 0.910483i \(-0.635710\pi\)
−0.413547 + 0.910483i \(0.635710\pi\)
\(422\) 0 0
\(423\) −9.65685 −0.469532
\(424\) 0 0
\(425\) −2.82843 −0.137199
\(426\) 0 0
\(427\) 3.31371 0.160362
\(428\) 0 0
\(429\) −23.3137 −1.12560
\(430\) 0 0
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) 0 0
\(433\) 12.6274 0.606835 0.303417 0.952858i \(-0.401872\pi\)
0.303417 + 0.952858i \(0.401872\pi\)
\(434\) 0 0
\(435\) 7.65685 0.367118
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −6.14214 −0.293148 −0.146574 0.989200i \(-0.546825\pi\)
−0.146574 + 0.989200i \(0.546825\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −6.34315 −0.301372 −0.150686 0.988582i \(-0.548148\pi\)
−0.150686 + 0.988582i \(0.548148\pi\)
\(444\) 0 0
\(445\) 15.6569 0.742206
\(446\) 0 0
\(447\) −0.343146 −0.0162302
\(448\) 0 0
\(449\) −21.3137 −1.00586 −0.502928 0.864328i \(-0.667744\pi\)
−0.502928 + 0.864328i \(0.667744\pi\)
\(450\) 0 0
\(451\) 36.9706 1.74088
\(452\) 0 0
\(453\) −8.48528 −0.398673
\(454\) 0 0
\(455\) −9.65685 −0.452720
\(456\) 0 0
\(457\) 18.9706 0.887405 0.443703 0.896174i \(-0.353665\pi\)
0.443703 + 0.896174i \(0.353665\pi\)
\(458\) 0 0
\(459\) −2.82843 −0.132020
\(460\) 0 0
\(461\) −8.34315 −0.388579 −0.194290 0.980944i \(-0.562240\pi\)
−0.194290 + 0.980944i \(0.562240\pi\)
\(462\) 0 0
\(463\) −20.6274 −0.958637 −0.479319 0.877641i \(-0.659116\pi\)
−0.479319 + 0.877641i \(0.659116\pi\)
\(464\) 0 0
\(465\) 6.82843 0.316661
\(466\) 0 0
\(467\) −35.3137 −1.63412 −0.817062 0.576550i \(-0.804399\pi\)
−0.817062 + 0.576550i \(0.804399\pi\)
\(468\) 0 0
\(469\) −3.31371 −0.153013
\(470\) 0 0
\(471\) 5.51472 0.254105
\(472\) 0 0
\(473\) −46.6274 −2.14393
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.343146 −0.0157116
\(478\) 0 0
\(479\) 21.6569 0.989527 0.494763 0.869028i \(-0.335255\pi\)
0.494763 + 0.869028i \(0.335255\pi\)
\(480\) 0 0
\(481\) −50.6274 −2.30841
\(482\) 0 0
\(483\) −11.3137 −0.514792
\(484\) 0 0
\(485\) 3.65685 0.166049
\(486\) 0 0
\(487\) −30.9706 −1.40341 −0.701705 0.712468i \(-0.747578\pi\)
−0.701705 + 0.712468i \(0.747578\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 32.8284 1.48153 0.740763 0.671766i \(-0.234464\pi\)
0.740763 + 0.671766i \(0.234464\pi\)
\(492\) 0 0
\(493\) 21.6569 0.975376
\(494\) 0 0
\(495\) −4.82843 −0.217022
\(496\) 0 0
\(497\) 4.68629 0.210209
\(498\) 0 0
\(499\) −38.6274 −1.72920 −0.864600 0.502460i \(-0.832428\pi\)
−0.864600 + 0.502460i \(0.832428\pi\)
\(500\) 0 0
\(501\) −10.3431 −0.462098
\(502\) 0 0
\(503\) −1.65685 −0.0738755 −0.0369377 0.999318i \(-0.511760\pi\)
−0.0369377 + 0.999318i \(0.511760\pi\)
\(504\) 0 0
\(505\) −17.3137 −0.770450
\(506\) 0 0
\(507\) 10.3137 0.458048
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 26.6274 1.17793
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 15.6569 0.689923
\(516\) 0 0
\(517\) −46.6274 −2.05067
\(518\) 0 0
\(519\) −15.6569 −0.687260
\(520\) 0 0
\(521\) 17.3137 0.758527 0.379264 0.925289i \(-0.376177\pi\)
0.379264 + 0.925289i \(0.376177\pi\)
\(522\) 0 0
\(523\) −29.9411 −1.30923 −0.654617 0.755961i \(-0.727170\pi\)
−0.654617 + 0.755961i \(0.727170\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 19.3137 0.841318
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) −0.828427 −0.0359507
\(532\) 0 0
\(533\) −36.9706 −1.60137
\(534\) 0 0
\(535\) 13.6569 0.590437
\(536\) 0 0
\(537\) 2.48528 0.107248
\(538\) 0 0
\(539\) −14.4853 −0.623925
\(540\) 0 0
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 0 0
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −14.3431 −0.613269 −0.306634 0.951827i \(-0.599203\pi\)
−0.306634 + 0.951827i \(0.599203\pi\)
\(548\) 0 0
\(549\) −1.65685 −0.0707128
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.97056 −0.381467
\(554\) 0 0
\(555\) −10.4853 −0.445075
\(556\) 0 0
\(557\) −12.3431 −0.522996 −0.261498 0.965204i \(-0.584216\pi\)
−0.261498 + 0.965204i \(0.584216\pi\)
\(558\) 0 0
\(559\) 46.6274 1.97213
\(560\) 0 0
\(561\) −13.6569 −0.576593
\(562\) 0 0
\(563\) −4.97056 −0.209484 −0.104742 0.994499i \(-0.533402\pi\)
−0.104742 + 0.994499i \(0.533402\pi\)
\(564\) 0 0
\(565\) 3.51472 0.147865
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 17.3137 0.725828 0.362914 0.931823i \(-0.381782\pi\)
0.362914 + 0.931823i \(0.381782\pi\)
\(570\) 0 0
\(571\) −6.62742 −0.277349 −0.138674 0.990338i \(-0.544284\pi\)
−0.138674 + 0.990338i \(0.544284\pi\)
\(572\) 0 0
\(573\) −27.3137 −1.14105
\(574\) 0 0
\(575\) 5.65685 0.235907
\(576\) 0 0
\(577\) −38.9706 −1.62237 −0.811183 0.584793i \(-0.801176\pi\)
−0.811183 + 0.584793i \(0.801176\pi\)
\(578\) 0 0
\(579\) 22.9706 0.954624
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.65685 −0.0686199
\(584\) 0 0
\(585\) 4.82843 0.199631
\(586\) 0 0
\(587\) −18.3431 −0.757103 −0.378551 0.925580i \(-0.623578\pi\)
−0.378551 + 0.925580i \(0.623578\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −21.3137 −0.876729
\(592\) 0 0
\(593\) −0.201010 −0.00825450 −0.00412725 0.999991i \(-0.501314\pi\)
−0.00412725 + 0.999991i \(0.501314\pi\)
\(594\) 0 0
\(595\) −5.65685 −0.231908
\(596\) 0 0
\(597\) −18.8284 −0.770596
\(598\) 0 0
\(599\) 34.3431 1.40322 0.701611 0.712560i \(-0.252464\pi\)
0.701611 + 0.712560i \(0.252464\pi\)
\(600\) 0 0
\(601\) 39.9411 1.62923 0.814616 0.580000i \(-0.196948\pi\)
0.814616 + 0.580000i \(0.196948\pi\)
\(602\) 0 0
\(603\) 1.65685 0.0674723
\(604\) 0 0
\(605\) −12.3137 −0.500623
\(606\) 0 0
\(607\) −20.6274 −0.837241 −0.418621 0.908161i \(-0.637486\pi\)
−0.418621 + 0.908161i \(0.637486\pi\)
\(608\) 0 0
\(609\) 15.3137 0.620543
\(610\) 0 0
\(611\) 46.6274 1.88634
\(612\) 0 0
\(613\) 26.4853 1.06973 0.534865 0.844937i \(-0.320362\pi\)
0.534865 + 0.844937i \(0.320362\pi\)
\(614\) 0 0
\(615\) −7.65685 −0.308754
\(616\) 0 0
\(617\) 2.82843 0.113868 0.0569341 0.998378i \(-0.481868\pi\)
0.0569341 + 0.998378i \(0.481868\pi\)
\(618\) 0 0
\(619\) −11.0294 −0.443311 −0.221655 0.975125i \(-0.571146\pi\)
−0.221655 + 0.975125i \(0.571146\pi\)
\(620\) 0 0
\(621\) 5.65685 0.227002
\(622\) 0 0
\(623\) 31.3137 1.25456
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −29.6569 −1.18250
\(630\) 0 0
\(631\) 36.4853 1.45246 0.726228 0.687454i \(-0.241272\pi\)
0.726228 + 0.687454i \(0.241272\pi\)
\(632\) 0 0
\(633\) 25.6569 1.01977
\(634\) 0 0
\(635\) 4.34315 0.172352
\(636\) 0 0
\(637\) 14.4853 0.573928
\(638\) 0 0
\(639\) −2.34315 −0.0926934
\(640\) 0 0
\(641\) 12.3431 0.487525 0.243762 0.969835i \(-0.421618\pi\)
0.243762 + 0.969835i \(0.421618\pi\)
\(642\) 0 0
\(643\) 2.62742 0.103615 0.0518076 0.998657i \(-0.483502\pi\)
0.0518076 + 0.998657i \(0.483502\pi\)
\(644\) 0 0
\(645\) 9.65685 0.380238
\(646\) 0 0
\(647\) −17.6569 −0.694163 −0.347081 0.937835i \(-0.612827\pi\)
−0.347081 + 0.937835i \(0.612827\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 13.6569 0.535254
\(652\) 0 0
\(653\) 29.3137 1.14713 0.573567 0.819159i \(-0.305559\pi\)
0.573567 + 0.819159i \(0.305559\pi\)
\(654\) 0 0
\(655\) 2.48528 0.0971080
\(656\) 0 0
\(657\) −13.3137 −0.519417
\(658\) 0 0
\(659\) −39.1716 −1.52591 −0.762954 0.646453i \(-0.776252\pi\)
−0.762954 + 0.646453i \(0.776252\pi\)
\(660\) 0 0
\(661\) 36.9706 1.43799 0.718994 0.695016i \(-0.244603\pi\)
0.718994 + 0.695016i \(0.244603\pi\)
\(662\) 0 0
\(663\) 13.6569 0.530388
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −43.3137 −1.67711
\(668\) 0 0
\(669\) 14.9706 0.578795
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 33.3137 1.28415 0.642075 0.766642i \(-0.278074\pi\)
0.642075 + 0.766642i \(0.278074\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 5.31371 0.204222 0.102111 0.994773i \(-0.467440\pi\)
0.102111 + 0.994773i \(0.467440\pi\)
\(678\) 0 0
\(679\) 7.31371 0.280674
\(680\) 0 0
\(681\) −6.34315 −0.243070
\(682\) 0 0
\(683\) 19.3137 0.739019 0.369509 0.929227i \(-0.379526\pi\)
0.369509 + 0.929227i \(0.379526\pi\)
\(684\) 0 0
\(685\) −6.82843 −0.260901
\(686\) 0 0
\(687\) 10.3431 0.394616
\(688\) 0 0
\(689\) 1.65685 0.0631211
\(690\) 0 0
\(691\) 38.6274 1.46946 0.734728 0.678362i \(-0.237310\pi\)
0.734728 + 0.678362i \(0.237310\pi\)
\(692\) 0 0
\(693\) −9.65685 −0.366834
\(694\) 0 0
\(695\) −1.65685 −0.0628481
\(696\) 0 0
\(697\) −21.6569 −0.820312
\(698\) 0 0
\(699\) −27.7990 −1.05145
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 9.65685 0.363698
\(706\) 0 0
\(707\) −34.6274 −1.30230
\(708\) 0 0
\(709\) 20.2843 0.761792 0.380896 0.924618i \(-0.375616\pi\)
0.380896 + 0.924618i \(0.375616\pi\)
\(710\) 0 0
\(711\) 4.48528 0.168211
\(712\) 0 0
\(713\) −38.6274 −1.44661
\(714\) 0 0
\(715\) 23.3137 0.871883
\(716\) 0 0
\(717\) −22.6274 −0.845036
\(718\) 0 0
\(719\) 44.2843 1.65152 0.825762 0.564018i \(-0.190745\pi\)
0.825762 + 0.564018i \(0.190745\pi\)
\(720\) 0 0
\(721\) 31.3137 1.16618
\(722\) 0 0
\(723\) −24.6274 −0.915903
\(724\) 0 0
\(725\) −7.65685 −0.284368
\(726\) 0 0
\(727\) 17.0294 0.631587 0.315793 0.948828i \(-0.397729\pi\)
0.315793 + 0.948828i \(0.397729\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 27.3137 1.01023
\(732\) 0 0
\(733\) −17.7990 −0.657421 −0.328710 0.944431i \(-0.606614\pi\)
−0.328710 + 0.944431i \(0.606614\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −25.9411 −0.954260 −0.477130 0.878833i \(-0.658323\pi\)
−0.477130 + 0.878833i \(0.658323\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.9706 1.06283 0.531413 0.847113i \(-0.321661\pi\)
0.531413 + 0.847113i \(0.321661\pi\)
\(744\) 0 0
\(745\) 0.343146 0.0125719
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27.3137 0.998021
\(750\) 0 0
\(751\) −5.45584 −0.199087 −0.0995433 0.995033i \(-0.531738\pi\)
−0.0995433 + 0.995033i \(0.531738\pi\)
\(752\) 0 0
\(753\) 3.17157 0.115579
\(754\) 0 0
\(755\) 8.48528 0.308811
\(756\) 0 0
\(757\) 17.5147 0.636583 0.318292 0.947993i \(-0.396891\pi\)
0.318292 + 0.947993i \(0.396891\pi\)
\(758\) 0 0
\(759\) 27.3137 0.991425
\(760\) 0 0
\(761\) 10.6863 0.387378 0.193689 0.981063i \(-0.437955\pi\)
0.193689 + 0.981063i \(0.437955\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 2.82843 0.102262
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) −9.31371 −0.335861 −0.167930 0.985799i \(-0.553708\pi\)
−0.167930 + 0.985799i \(0.553708\pi\)
\(770\) 0 0
\(771\) 9.17157 0.330306
\(772\) 0 0
\(773\) 51.2548 1.84351 0.921754 0.387775i \(-0.126756\pi\)
0.921754 + 0.387775i \(0.126756\pi\)
\(774\) 0 0
\(775\) −6.82843 −0.245284
\(776\) 0 0
\(777\) −20.9706 −0.752315
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) 0 0
\(783\) −7.65685 −0.273634
\(784\) 0 0
\(785\) −5.51472 −0.196829
\(786\) 0 0
\(787\) −39.3137 −1.40138 −0.700691 0.713465i \(-0.747125\pi\)
−0.700691 + 0.713465i \(0.747125\pi\)
\(788\) 0 0
\(789\) −18.3431 −0.653033
\(790\) 0 0
\(791\) 7.02944 0.249938
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 0.343146 0.0121701
\(796\) 0 0
\(797\) −10.9706 −0.388597 −0.194299 0.980942i \(-0.562243\pi\)
−0.194299 + 0.980942i \(0.562243\pi\)
\(798\) 0 0
\(799\) 27.3137 0.966290
\(800\) 0 0
\(801\) −15.6569 −0.553208
\(802\) 0 0
\(803\) −64.2843 −2.26854
\(804\) 0 0
\(805\) 11.3137 0.398756
\(806\) 0 0
\(807\) 3.65685 0.128727
\(808\) 0 0
\(809\) 12.3431 0.433962 0.216981 0.976176i \(-0.430379\pi\)
0.216981 + 0.976176i \(0.430379\pi\)
\(810\) 0 0
\(811\) −43.5980 −1.53093 −0.765466 0.643476i \(-0.777492\pi\)
−0.765466 + 0.643476i \(0.777492\pi\)
\(812\) 0 0
\(813\) 8.48528 0.297592
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 9.65685 0.337438
\(820\) 0 0
\(821\) −1.02944 −0.0359276 −0.0179638 0.999839i \(-0.505718\pi\)
−0.0179638 + 0.999839i \(0.505718\pi\)
\(822\) 0 0
\(823\) 38.2843 1.33451 0.667253 0.744831i \(-0.267470\pi\)
0.667253 + 0.744831i \(0.267470\pi\)
\(824\) 0 0
\(825\) 4.82843 0.168104
\(826\) 0 0
\(827\) −16.6863 −0.580239 −0.290120 0.956990i \(-0.593695\pi\)
−0.290120 + 0.956990i \(0.593695\pi\)
\(828\) 0 0
\(829\) −29.9411 −1.03990 −0.519949 0.854197i \(-0.674049\pi\)
−0.519949 + 0.854197i \(0.674049\pi\)
\(830\) 0 0
\(831\) 28.8284 1.00005
\(832\) 0 0
\(833\) 8.48528 0.293998
\(834\) 0 0
\(835\) 10.3431 0.357939
\(836\) 0 0
\(837\) −6.82843 −0.236025
\(838\) 0 0
\(839\) −44.2843 −1.52886 −0.764431 0.644705i \(-0.776980\pi\)
−0.764431 + 0.644705i \(0.776980\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 0 0
\(843\) 22.9706 0.791148
\(844\) 0 0
\(845\) −10.3137 −0.354802
\(846\) 0 0
\(847\) −24.6274 −0.846208
\(848\) 0 0
\(849\) −20.9706 −0.719708
\(850\) 0 0
\(851\) 59.3137 2.03325
\(852\) 0 0
\(853\) 18.4853 0.632924 0.316462 0.948605i \(-0.397505\pi\)
0.316462 + 0.948605i \(0.397505\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.14214 −0.209811 −0.104906 0.994482i \(-0.533454\pi\)
−0.104906 + 0.994482i \(0.533454\pi\)
\(858\) 0 0
\(859\) 54.9117 1.87356 0.936781 0.349915i \(-0.113790\pi\)
0.936781 + 0.349915i \(0.113790\pi\)
\(860\) 0 0
\(861\) −15.3137 −0.521890
\(862\) 0 0
\(863\) 10.3431 0.352085 0.176042 0.984383i \(-0.443670\pi\)
0.176042 + 0.984383i \(0.443670\pi\)
\(864\) 0 0
\(865\) 15.6569 0.532349
\(866\) 0 0
\(867\) −9.00000 −0.305656
\(868\) 0 0
\(869\) 21.6569 0.734658
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −3.65685 −0.123766
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −2.48528 −0.0839220 −0.0419610 0.999119i \(-0.513361\pi\)
−0.0419610 + 0.999119i \(0.513361\pi\)
\(878\) 0 0
\(879\) −5.31371 −0.179227
\(880\) 0 0
\(881\) 35.6569 1.20131 0.600655 0.799508i \(-0.294907\pi\)
0.600655 + 0.799508i \(0.294907\pi\)
\(882\) 0 0
\(883\) 37.9411 1.27682 0.638410 0.769696i \(-0.279592\pi\)
0.638410 + 0.769696i \(0.279592\pi\)
\(884\) 0 0
\(885\) 0.828427 0.0278473
\(886\) 0 0
\(887\) −39.5980 −1.32957 −0.664785 0.747035i \(-0.731477\pi\)
−0.664785 + 0.747035i \(0.731477\pi\)
\(888\) 0 0
\(889\) 8.68629 0.291329
\(890\) 0 0
\(891\) 4.82843 0.161758
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −2.48528 −0.0830738
\(896\) 0 0
\(897\) −27.3137 −0.911978
\(898\) 0 0
\(899\) 52.2843 1.74378
\(900\) 0 0
\(901\) 0.970563 0.0323341
\(902\) 0 0
\(903\) 19.3137 0.642720
\(904\) 0 0
\(905\) −12.0000 −0.398893
\(906\) 0 0
\(907\) 12.9706 0.430680 0.215340 0.976539i \(-0.430914\pi\)
0.215340 + 0.976539i \(0.430914\pi\)
\(908\) 0 0
\(909\) 17.3137 0.574259
\(910\) 0 0
\(911\) 35.3137 1.17000 0.584998 0.811035i \(-0.301095\pi\)
0.584998 + 0.811035i \(0.301095\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1.65685 0.0547739
\(916\) 0 0
\(917\) 4.97056 0.164142
\(918\) 0 0
\(919\) 44.4853 1.46743 0.733717 0.679455i \(-0.237784\pi\)
0.733717 + 0.679455i \(0.237784\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) 10.4853 0.344754
\(926\) 0 0
\(927\) −15.6569 −0.514239
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 11.3137 0.370394
\(934\) 0 0
\(935\) 13.6569 0.446627
\(936\) 0 0
\(937\) 13.3137 0.434940 0.217470 0.976067i \(-0.430220\pi\)
0.217470 + 0.976067i \(0.430220\pi\)
\(938\) 0 0
\(939\) −10.9706 −0.358011
\(940\) 0 0
\(941\) −35.2548 −1.14927 −0.574637 0.818408i \(-0.694857\pi\)
−0.574637 + 0.818408i \(0.694857\pi\)
\(942\) 0 0
\(943\) 43.3137 1.41049
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 45.9411 1.49289 0.746443 0.665449i \(-0.231760\pi\)
0.746443 + 0.665449i \(0.231760\pi\)
\(948\) 0 0
\(949\) 64.2843 2.08676
\(950\) 0 0
\(951\) 9.31371 0.302018
\(952\) 0 0
\(953\) −33.1716 −1.07453 −0.537266 0.843413i \(-0.680543\pi\)
−0.537266 + 0.843413i \(0.680543\pi\)
\(954\) 0 0
\(955\) 27.3137 0.883851
\(956\) 0 0
\(957\) −36.9706 −1.19509
\(958\) 0 0
\(959\) −13.6569 −0.441003
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) −13.6569 −0.440086
\(964\) 0 0
\(965\) −22.9706 −0.739449
\(966\) 0 0
\(967\) 56.9117 1.83016 0.915078 0.403276i \(-0.132129\pi\)
0.915078 + 0.403276i \(0.132129\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.8579 0.508903 0.254452 0.967086i \(-0.418105\pi\)
0.254452 + 0.967086i \(0.418105\pi\)
\(972\) 0 0
\(973\) −3.31371 −0.106233
\(974\) 0 0
\(975\) −4.82843 −0.154633
\(976\) 0 0
\(977\) −26.8284 −0.858317 −0.429159 0.903229i \(-0.641190\pi\)
−0.429159 + 0.903229i \(0.641190\pi\)
\(978\) 0 0
\(979\) −75.5980 −2.41612
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 19.0294 0.606945 0.303472 0.952840i \(-0.401854\pi\)
0.303472 + 0.952840i \(0.401854\pi\)
\(984\) 0 0
\(985\) 21.3137 0.679111
\(986\) 0 0
\(987\) 19.3137 0.614762
\(988\) 0 0
\(989\) −54.6274 −1.73705
\(990\) 0 0
\(991\) 15.5147 0.492841 0.246421 0.969163i \(-0.420746\pi\)
0.246421 + 0.969163i \(0.420746\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) 18.8284 0.596901
\(996\) 0 0
\(997\) 12.1421 0.384545 0.192273 0.981342i \(-0.438414\pi\)
0.192273 + 0.981342i \(0.438414\pi\)
\(998\) 0 0
\(999\) 10.4853 0.331740
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3840.2.a.bj.1.2 2
4.3 odd 2 3840.2.a.bd.1.1 2
8.3 odd 2 3840.2.a.bm.1.2 2
8.5 even 2 3840.2.a.bg.1.1 2
16.3 odd 4 1920.2.k.j.961.2 4
16.5 even 4 1920.2.k.k.961.2 yes 4
16.11 odd 4 1920.2.k.j.961.3 yes 4
16.13 even 4 1920.2.k.k.961.3 yes 4
48.5 odd 4 5760.2.k.x.2881.3 4
48.11 even 4 5760.2.k.m.2881.4 4
48.29 odd 4 5760.2.k.x.2881.2 4
48.35 even 4 5760.2.k.m.2881.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1920.2.k.j.961.2 4 16.3 odd 4
1920.2.k.j.961.3 yes 4 16.11 odd 4
1920.2.k.k.961.2 yes 4 16.5 even 4
1920.2.k.k.961.3 yes 4 16.13 even 4
3840.2.a.bd.1.1 2 4.3 odd 2
3840.2.a.bg.1.1 2 8.5 even 2
3840.2.a.bj.1.2 2 1.1 even 1 trivial
3840.2.a.bm.1.2 2 8.3 odd 2
5760.2.k.m.2881.1 4 48.35 even 4
5760.2.k.m.2881.4 4 48.11 even 4
5760.2.k.x.2881.2 4 48.29 odd 4
5760.2.k.x.2881.3 4 48.5 odd 4